Finite Group Databasebeta

Subgroups — #14529381

Order 2176.

186 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762845	C₂	1	1	yes	maximal, characteristic
2	#11762837	C₂	1	1	yes	maximal, characteristic
3	#11762765	C₂	1	1	yes	maximal, characteristic
4	#11762573	C₂	1	1	yes	maximal, characteristic
5	#11762569	C₂	1	1	yes	maximal, characteristic
6	#11762564	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	#10587612	C₂²	1	1	yes	characteristic
12	C₂×#63223	C₂²	1	1	yes	characteristic
13	C₂²×#3622	C₂²	1	1	yes	characteristic
14	C₂²×#3622	C₂²	1	1	yes	characteristic
15	#10587485	—	2	1	no	—
16	#10587485	—	2	1	no	—
17	#10587479	C₄	1	1	yes	characteristic, complemented
18	#10587479	C₄	1	1	yes	characteristic, complemented
19	#10587478	C₄	1	1	yes	characteristic, complemented
20	#10587478	C₄	1	1	yes	characteristic, complemented
21	#10587474	—	2	1	no	—
22	C₄×#3622	—	2	1	no	—
23	C₂²×C₄×C₁₇	—	2	1	no	—
24	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₇	—	2	1	no	—
26	M₁₆×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	—	2	1	no	—
28	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²⋊C₄×C₁₇	—	2	1	no	—
31	C₄²×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	—	2	1	no	—
33	C₄²×C₁₇	—	2	1	no	—
34	C₂²⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₇	—	2	1	no	—
38	#63223	—	2	1	no	—
39	#63223	Q₈	1	2	yes	—
40	C₂×#3622	—	2	1	no	—
41	C₂×#3622	—	2	1	no	—
42	C₂×#3622	—	2	1	no	—
43	C₂×#3622	—	2	1	no	—
44	C₂×#3622	—	2	1	no	—
45	C₂×#3622	C₂×C₄	1	1	yes	characteristic
46	C₂×#3622	C₂×C₄	1	1	yes	characteristic
47	C₂×#3622	D₄	1	1	yes	characteristic
48	C₂×#3622	D₄	1	1	yes	characteristic
49	C₂³×C₁₇	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	—	2	1	no	—
52	C₂×C₄×C₁₇	—	2	1	no	—
53	C₂×C₄×C₁₇	—	4	1	no	—
54	C₂×C₄×C₁₇	—	4	1	no	—
55	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
61	C₂×Dic₁₇	—	2	1	no	—
62	C₂×Dic₁₇	—	2	1	no	—
63	C₈×C₁₇	—	4	1	no	—
64	#3622	—	2	1	no	—
65	#3622	—	2	1	no	—
66	#3622	—	2	2	no	—
67	#3622	—	2	1	no	—
68	#3622	—	2	1	no	—
69	#1922	—	17	1	no	maximal, Sylow
70	C₂²×C₁₇	—	2	1	no	—
71	C₂²×C₁₇	—	2	1	no	—
72	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₁₇	#125	1	1	yes	characteristic
74	C₂²×C₁₇	#126	1	1	yes	characteristic
75	C₄×C₁₇	—	4	2	no	—
76	C₄×C₁₇	—	4	1	no	—
77	C₄×C₁₇	—	4	1	no	—
78	C₄×C₁₇	#117	1	1	yes	characteristic
79	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
80	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₁₇	#123	1	1	yes	characteristic
82	Dic₁₇	—	4	1	no	—
83	Dic₁₇	—	4	1	no	—
84	#427	—	17	1	no	—
85	#424	—	17	1	no	—
86	#405	—	17	1	no	—
87	#341	—	17	1	no	—
88	#339	—	17	1	no	—
89	#339	D₁₇	1	1	yes	characteristic
90	#336	—	17	1	no	—
91	C₂×C₁₇	—	2	1	no	—
92	C₂×C₁₇	—	2	1	no	—
93	C₂×C₁₇	#387	1	1	yes	characteristic
94	C₂×C₁₇	#443	1	1	yes	characteristic
95	C₂×C₁₇	#444	1	1	yes	characteristic
96	C₂×M₁₆	—	17	1	no	—
97	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
98	C₂²×C₈	—	17	1	no	—
99	C₂²×C₈	—	17	1	no	—
100	#117	—	17	1	no	—
101	#117	C₂×D₁₇	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
103	#107	—	17	1	no	—
104	#107	—	17	1	no	—
105	#106	—	17	1	no	—
106	#106	—	17	1	no	—
107	C₄⋊C₈	—	34	1	no	—
108	C₄⋊C₈	—	34	1	no	—
109	#97	—	34	1	no	—
110	C₄×C₈	—	34	1	no	—
111	C₁₇	#1922	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	17	2	no	—
117	M₁₆	—	34	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	34	1	no	—
124	C₂×C₈	—	34	1	no	—
125	C₂×C₈	—	34	1	no	—
126	C₂×C₈	—	34	1	no	—
127	C₂×C₈	—	34	1	no	—
128	C₄⋊C₄	—	17	1	no	—
129	C₄⋊C₄	—	17	1	no	—
130	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
131	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	34	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	34	1	no	—
138	C₂³	#63254	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	34	1	no	—
145	C₂×C₄	—	34	1	no	—
146	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
147	C₂×C₄	#63252	1	1	yes	characteristic
148	C₂×C₄	#63252	1	1	yes	characteristic
149	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
150	C₂×C₄	#63254	1	1	yes	characteristic
151	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
152	C₈	—	4	1	no	—
153	C₈	—	34	1	no	—
154	C₈	—	34	1	no	—
155	C₈	—	34	1	no	—
156	C₈	—	34	1	no	—
157	C₈	—	34	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#10587565	1	1	yes	characteristic
161	C₂²	#10587567	1	1	yes	characteristic
162	C₂²	#10587575	1	1	yes	characteristic
163	C₄	—	4	2	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	68	1	no	—
167	C₄	—	68	1	no	—
168	C₄	#10587558	1	1	yes	characteristic
169	C₄	#10587569	1	1	yes	characteristic
170	C₄	#10587573	1	1	yes	characteristic
171	C₄	#10587582	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#11762691	1	1	yes	characteristic
175	C₂	#11762923	1	1	yes	characteristic
176	C₂	#11762928	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#339	C₂
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#339	D₁₇
hypocenter	C₁₇	#1922
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
17. #10587479	C₄	semidirect
18. #10587479	C₄	semidirect
19. #10587478	C₄	semidirect
20. #10587478	C₄	semidirect
111. C₁₇	#1922	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11762845	2
#11762837	2
#11762765	2
#11762573	2
#11762569	2
#11762564	2
C₁₇×#339	2
#1922	17